Bevel Tool on two combined rectangles
Here's one way:

Scale a cube to a shape like in the first picture. Then cut a loop using the Loop Cut and Slide tool. (Picture 2.) Change the selection mode to Faces and select the end Face shown in picture 3. Press E to extrude, then drag away from the face to "pull out" a new section. Now you can use your bevel tool on the shape and there'll be no gap.